Chapter 7.4 — Exercise 7.3 — Graphs

The Histogram

A histogram draws each class interval as a bar sitting directly on the x-axis, with the bar's height equal to that class's frequency and its width spanning the class interval exactly — bars for consecutive classes touch, with no gap between them, since the classes themselves are continuous. For 45 students grouped by IQ into seven classes (60–70 through 120–130) with frequencies 2, 5, 6, 10, 9, 8, 5, the tallest bar sits over 90–100 (frequency 10), immediately showing where most students fall without reading a single number off the table.

Choosing a sensible scale for each axis matters as much as the bars themselves — marking the IQ axis in steps of 10 and the count axis in steps of 1 keeps the histogram readable; a scale mismatched to the data either compresses everything into an unreadably thin sliver or stretches it across more space than the page allows.

The Frequency Polygon

Rather than bars, a frequency polygon plots a single point at each class's midpoint (its class mark), at a height equal to that class's frequency, then joins the points with straight line segments. For weekly wages grouped into six classes with class marks 525, 575, 625, 675, 725, 775 and frequencies 30, 42, 50, 55, 45, 28, the plotted points (525,30), (575,42), (625,50), (675,55), (725,45), (775,28) joined in sequence trace out the same rise-then-fall shape a histogram's bar heights would show, just as a connected line instead of blocks.

A frequency polygon can be drawn directly on top of a histogram (joining the midpoint of each bar's top edge) or built independently straight from the class marks and frequencies — both approaches land on exactly the same set of points.

Why Bars Touch Instead of Standing Apart

It's worth being deliberate about why a histogram's bars touch, unlike an ordinary bar chart comparing separate categories (favourite colours, say), where gaps between bars are normal and expected. A histogram's x-axis represents a continuous quantity — IQ, age, marks — where every value between one class's end and the next class's start genuinely exists and belongs to one class or the other. Leaving a gap between bars would visually suggest a range of values that belongs to no class at all, which misrepresents data that is, in reality, continuous. This single distinction — gaps allowed for categories, gaps forbidden for continuous class intervals — is what separates a histogram from an ordinary bar chart, even though the two look superficially similar.

The Frequency Curve

A frequency curve is a frequency polygon's points joined with a smooth, continuous curve instead of straight line segments — useful when the underlying quantity (like age) is thought of as varying continuously rather than jumping between discrete class marks. For 60 teachers' ages grouped into six classes with class marks 26, 30, 34, 38, 42, 46 and frequencies 12, 10, 15, 9, 8, 6, the same six points used for a frequency polygon become the basis for a smoothed curve instead — the data doesn't change, only how the points between them are connected.

Why a Polygon's Ends Need Extra, Empty Classes

A detail easy to overlook: a properly drawn frequency polygon doesn't just stop at the first and last class marks — it's brought down to the x-axis at an extra class mark on either end, one imagined empty class before the first real one and one after the last, each with frequency zero. Without this, the polygon would end abruptly mid-air above the axis rather than closing off cleanly at zero, which matters because a genuine polygon (a closed shape) is what makes the area under it meaningful for comparison against other data sets drawn the same way. This is a small construction detail, but skipping it is one of the more common ways a frequency polygon ends up drawn incorrectly.

The Ogive — Graphing a Cumulative Frequency

An ogive plots cumulative frequency (rather than plain frequency) against the class boundaries, producing a curve that only ever rises (for a less-than ogive) or only ever falls (for a greater-than ogive) — never both, since a running total can't decrease as more classes are included.

For less-than cumulative frequencies of 2, 8, 18, 27, 35 across classes with upper boundaries 5, 10, 15, 20, 25: plotting (5,2), (10,8), (15,18), (20,27), (25,35) and joining them traces a curve that climbs steadily as the boundary increases — a less-than ogive. The corresponding greater-than cumulative frequencies (35, 33, 27, 17, 8) plotted against the classes' lower boundaries (0, 5, 10, 15, 20) trace a curve that steadily falls instead — a greater-than ogive.

Drawing both ogives on the same graph has a genuinely useful payoff beyond this exercise: the x-coordinate where the rising less-than curve crosses the falling greater-than curve gives the median of the grouped data directly, without any of the class-interval arithmetic from the median formula — the two cumulative counts are exactly equal at that point, which is precisely what the median represents — the value below which exactly half the data sits, and above which the other half sits, found by reading a single intersection point rather than working through the class-interval median formula by hand.

Plotting Points Accurately Before Joining Them

Every graph type in this exercise starts the same way underneath the surface: identify the correct pair of coordinates for each data point, mark them precisely against the chosen scale, and only then connect them according to whichever graph is being built — bars for a histogram, straight segments for a polygon, a smooth curve for a frequency curve, a rising or falling curve for an ogive. Since the connecting step differs across all four, but the plotting step is identical every time, most construction errors actually trace back to a misread coordinate or a misjudged scale, not to the joining method itself.

Matching the Graph Type to the Question Being Asked

Each of the four graph types answers a slightly different question at a glance: a histogram shows which class has the most observations; a frequency polygon or curve shows the overall shape of the distribution across all classes at once; an ogive shows how many observations fall below (or above) any given point, including points that don't align with a class boundary at all. Recognising which of these questions a problem is actually asking determines which graph is worth constructing — building a histogram when an ogive's cumulative reading is what's needed wastes the effort of a graph that doesn't answer the question in front of you.

Where This Leads

The grouped-data foundations this exercise turns into pictures come from Exercise 7.2, and the same statistical graphing ideas are developed further, alongside more formal measures of spread, in Statistics in Class 9 and again in Class 10, where ogive curves are used routinely to find the median of large data sets.